# LMIs in Control/pages/Stability of Lure systems

## The System

{\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{p}p(t)+B_{w}w(t),\\z(t)&=C_{z}x(t)\\p_{i}(t)&=\phi _{i}(q_{i}(t)),i=1,\dots ,n_{p}\\q&=C_{q}x,\\0&\leq \sigma \phi _{i}(\sigma )\leq \sigma ^{2}\ \forall \sigma \in \mathbb {R} \end{aligned}}

## The Data

The matrices $A,B_{p},B_{w},C_{q},C_{z}$ .

## The LMI: The Lure's System's Stability

The following feasibility problem should be solved as sufficient condition for the stability of the above Lur'e system.

{\begin{aligned}{\text{Find}}\;&P>0,\Lambda =diag(\lambda _{1},\dots ,\lambda _{n_{p}})\succeq 0,T=diag(\tau _{1},\dots ,\tau _{n_{p}})\succeq 0:\\&{\begin{bmatrix}A^{\top }P+PA&PB_{p}+A^{\top }C_{q}^{\top }\Lambda +C_{q}^{\top }T\\B_{p}^{\top }P+\Lambda C_{q}A+TC_{q}&\Lambda C_{q}B_{p}+B_{p}^{\top }C_{q}^{\top }\Lambda -2T\end{bmatrix}}\prec 0\\\end{aligned}}

## Conclusion

If the feasibility problem with LMI constraints has solution, then the Lure's system is stable.

## Remark

The LMI is only a sufficient condition for the existence of a Lur’e Lyapunov function that proves stability of Lur'e system . It is also necessary when there is only one nonlinearity, i.e., when $n_{p}=1$ .